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The bead model and limit behaviors of dimer models

Cédric Boutillier
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Abstract

In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire; there must be exactly one bead on each neighboring wire. We construct a one-parameter family of Gibbs measures on the bead configurations that are uniform in a certain sense. When endowed with one of these measures, this model is shown to be a determinantal point process, whose marginal on each wire is the sine process (given by eigenvalues of large hermitian random matrices). We prove then that this process appears as a limit of any dimer model on a planar bipartite graph when some weights degenerate.

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Cédric Boutillier. 2006-07-06. The bead model and limit behaviors of dimer models. https://doi.org/10.1214/08-aop398

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